More information about Deidara can be found on this link:
When Deidara's exploits caught the attention of the Akatsuki, Itachi
Uchiha, Kisame Hoshigaki, and Sasori were sent to recruit him. Deidara
refused at first but accepts a challenge from Itachi on the condition
that the Akatsuki let him be if he won. However, Deidara was easily
defeated by Itachi's Sharingan and begrudgingly joined the
organisation. -> Chapter 359.
His ego crushed since then because he admitted to himself that the
Sharingan was a work of art, Deidara carried a murderous loathing for
Itachi and the Sharingan for the rest of his life. Acknowledging that
Itachi was a more powerful opponent, Deidara took steps during his
time with the Akatsuki to get revenge on Itachi, such as training his
left eye to counter genjutsu, and devising the means to create C4.
Now since Itachi easily defeated Orochimaru, Deidara wanted to defeat
him as well to show off and prove to Itachi that he is strong. You can
say, Itachi defeated Orochimaru, then he MUST defeat him as well.